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FRM Exam Part I · Measuring Return, Volatility, and Correlation

Implied Volatility and the VIX Index Explained

Updated 11 October 2026 · Fact-checked

Implied volatility is the volatility value that, put into an option pricing model such as Black-Scholes-Merton, returns the option's observed market price. It is forward-looking, unlike historical volatility. The VIX applies a model-free formula to S&P 500 option prices to estimate expected 30-day volatility, quoted in annualized percentage points.

Understand Implied Volatility and VIX

Historical volatility is the standard deviation of past returns, scaled to a year. It looks backward. It tells you how much the price moved, not how much the market expects it to move.

Implied volatility (IV) works the other way round. An option's price depends on the stock price, strike, time, interest rate, dividends and volatility. All of these are observable except volatility. So you take the market price and find the volatility that makes the model price equal it. There is no closed-form inverse for Black-Scholes-Merton, so you solve it numerically, for example by trial and error or Newton-Raphson. Call price rises with volatility, so exactly one IV matches a given price.

IV is the market's view of future volatility, but it also contains a risk premium and supply and demand effects. It is often higher than later realized volatility. Equity options also show a volatility skew or smile: options at different strikes imply different volatilities. This contradicts the constant-volatility assumption of Black-Scholes-Merton. Quoting prices as IV is simply a convenient way to compare options across strikes and maturities.

The VIX is an index of expected 30-day volatility of the S&P 500. It is built from a strip of out-of-the-money calls and puts on the index, across many strikes. Each option is weighted by its price and by the strike spacing, divided by the strike squared. This gives an estimate of the risk-neutral variance, which is why VIX is called model-free: it needs no Black-Scholes-Merton assumption. The result is quoted as an annualized percentage. A VIX of 20 means the market-implied annualized volatility is about 20%.

A useful link: a variance swap's fair strike is based on the same calculation, so VIX squared (in percentage terms) approximates the fair variance swap rate.

Key formulas to remember

Implied volatility definition
Find σ such that Model(S, K, T, r, q, σ) = Market price
Solved numerically. Only volatility is unknown, so one price gives one IV.
Annualizing and scaling volatility
σ(T days) = σ(annual) × √(T ÷ 252)
Uses the square-root-of-time rule. Use the trading-day count the question gives, commonly 252.
VIX variance formula
σ² = (2 ÷ T) Σ [ΔKᵢ ÷ Kᵢ²] e^(RT) Q(Kᵢ) − (1 ÷ T)[F ÷ K₀ − 1]²
Q(Kᵢ) is the midpoint of the bid-ask of the out-of-the-money option at strike Kᵢ. K₀ is the first strike below the forward F. VIX = 100 × σ.
Approximate VIX with a variance swap rate
VIX ≈ 100 × √(expected average variance over 30 days, annualized)
Fair variance swap strike is the risk-neutral expected variance.
Vega sign
Vega > 0 for long calls and long puts
Higher IV raises option prices, so IV rising helps long option positions.

How to solve Implied Volatility and VIX questions

Use this sequence for most questions on implied volatility, historical volatility and VIX.

  1. 1Identify what is asked: a definition, a comparison, a calculation of IV, a scaling of volatility, or an interpretation of VIX.
  2. 2Check whether the number is annualized or for a shorter period. VIX and quoted IVs are annualized.
  3. 3For IV questions, note that the model price is increasing in σ. Compare the model price at a trial σ with the market price and move σ up or down.
  4. 4Convert between horizons with the square-root-of-time rule: multiply annual volatility by √(days ÷ 252) for a daily figure, or the reverse.
  5. 5For VIX, remember it uses out-of-the-money options across strikes, is model-free, and measures 30-day expected volatility of the S&P 500.
  6. 6For a comparison, state direction: IV is forward-looking and market-based; historical volatility is backward-looking and data-based.
  7. 7Sanity-check the answer: percentages should be plausible and the sign of any vega effect should match the position.

Quickest way: Interpolate and scale

When to use it: Use when the question gives two trial volatilities with model prices around the market price, or asks you to convert VIX to a daily move.

  1. For IV: if the market price lies between two model prices at σ₁ and σ₂, interpolate linearly: σ ≈ σ₁ + (P − P₁) ÷ (P₂ − P₁) × (σ₂ − σ₁).
  2. For a one-day move from VIX: divide VIX by √252 (about 15.87).
  3. For a one-standard-deviation move over n days: VIX ÷ 100 × √(n ÷ 252).
  4. Eliminate options that say VIX is based on historical data or on one at-the-money option.

Common mistakes in Implied Volatility and VIX

  • Treating implied volatility as a forecast guaranteed to come true.

    It is called forward-looking, so students assume it is unbiased.

    Fix: IV reflects market expectations plus a risk premium and demand effects. It often exceeds later realized volatility.

  • Saying VIX is calculated with the Black-Scholes-Merton formula.

    Students link all implied volatility to Black-Scholes.

    Fix: VIX is model-free. It weights out-of-the-money option prices across many strikes.

  • Forgetting that VIX is annualized.

    It is described as 30-day, so students read it as a 30-day volatility.

    Fix: The horizon is 30 days, but the number is quoted per year. Scale by √(T) for other horizons.

  • Using only at-the-money options for VIX.

    At-the-money options are the most liquid and familiar.

    Fix: VIX uses out-of-the-money puts below the forward and out-of-the-money calls above it.

  • Assuming IV is the same across strikes.

    Black-Scholes-Merton assumes constant volatility.

    Fix: Observed IVs form a skew or smile. Equity index options typically show higher IV for low strikes.

  • Using 365 days when the question implies 252 trading days.

    Mixing calendar and trading day conventions.

    Fix: Use the convention stated in the question. Default to 252 for volatility scaling.

Worked examples

Example 1

The VIX closes at 25. Using 252 trading days, estimate the one-standard-deviation daily move in the S&P 500, in percent.

Show the solution
  1. VIX is annualized volatility in percent: σ = 25%.
  2. Daily volatility = 25% × √(1 ÷ 252).
  3. √252 ≈ 15.875.
  4. Daily volatility = 25 ÷ 15.875 ≈ 1.575%.

Answer: About 1.57% per day.

Example 2

A call option trades at 4.20. The Black-Scholes-Merton price is 3.90 at σ = 20% and 4.50 at σ = 24%. Using linear interpolation, estimate the implied volatility.

Show the solution
  1. Price rises with volatility, so IV lies between 20% and 24%.
  2. Fraction of the gap: (4.20 − 3.90) ÷ (4.50 − 3.90) = 0.30 ÷ 0.60 = 0.5.
  3. IV ≈ 20% + 0.5 × (24% − 20%) = 20% + 2% = 22%.

Answer: Implied volatility ≈ 22%.

Exam tips

  • Expect conceptual questions: forward-looking versus backward-looking, and what VIX measures. Learn the key words: model-free, out-of-the-money, 30-day, annualized.
  • If a question asks which option's price rises when IV rises, the answer is a long call or long put, since vega is positive.
  • Be ready to scale an annual volatility to a daily or monthly figure with the square-root rule.
  • A smile or skew is evidence against constant volatility in Black-Scholes-Merton. Link it to fat tails in returns.
  • You do not need to compute the full VIX sum by hand. Know its structure and what each term does.

Practice questions from Measuring Return, Volatility, and Correlation

Implied Volatility and VIX in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Implied Volatility and VIX: frequently asked questions

What is the difference between implied and historical volatility?

Historical volatility is the annualized standard deviation of past returns. Implied volatility is backed out of current option prices and reflects what the market expects. One looks back, the other looks forward.

How is the VIX calculated?

It uses prices of out-of-the-money S&P 500 calls and puts across many strikes, with near-term and next-term expiries combined to give a constant 30-day horizon. Each option is weighted by strike spacing divided by strike squared, and the sum gives expected variance. VIX is 100 times the square root of that variance.

Can implied volatility be calculated with a formula?

Not in closed form for Black-Scholes-Merton. You solve for it numerically by adjusting volatility until the model price equals the market price. Calculators and software do this iteration for you.

Does a high VIX mean the market will fall?

No. A high VIX means the market expects large price swings and is paying more for options protection. It signals uncertainty, not direction.